Differential Equations Flashcards
What is the general form for a second order homogeneous differential equation?
ay’’ + by’ + cy = 0
What is the auxillary equation for a second order differential equation?
am^2 + bm + c = 0
What is the general solution to a second order homogenous differential equation (with real roots of the auxillary equation)?
y = A exp(ax) + B exp(bx)
where a and b are the solutions to the auxillary equation and A and B are arbitrary constants
What is the general solution to a second order homogenous differential equation (with complex roots [ a +- bi ] of the auxillary equation)?
y = exp(ax) [ A cos(bx) + B sin(bx) ]
What is the general solution to a homogenous second order differential equation where the auxillary equation has a repeated root?
y = (A + Bx) exp(ax)
where a is the repeated root and A, B are arbitrary constants
What is the general formula of a non homogeneous second order differential equation?
ay’’ + by’ + c = f(x)
where f(x) isn’t 0
f(x) = k
Particular integral = ?
λ
(a constant)
f(x) = ax + b
Particular integral = ?
λ + µx
f(x) = ax^2 + bx + c
Particular Integral = ?
λ + µx + vx^2
f(x) = k exp( px )
Particular integral = ?
λ exp( px )
where p is the same as in f(x)
f(x) = m cos(ωx)
particular integral = ?
λ cos(ωx) + µ sin(ωx)
f(x) = m sin(ωx)
particular integral = ?
λ cos(ωx) + µ sin(ωx)
f(x) = m cos(ωx) + n sin(ωx)
particular integral = ?
λ cos(ωx) + µ sin(ωx)
What is the general solution to a non homogenous second order differential equation?
y = complimentary function (CF) + particular integral (PI)
CF = general solution if it was homogenous
PI determined by f(x)
